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Proof-Theoretic Semantics, a Problem with Negation and Prospects for Modality

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Abstract

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings of modal operators in terms of rules of inference.

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Notes

  1. [22–28] is a representative selection of Prawitz’ papers.

  2. This is an interesting omission: in the systems of Gentzen [16], structural rules are as natural a part of natural deduction as inference rules.

  3. It is sometimes said that there are inconsistent, i.e. non-conservative, connectives that have stable rules. This is impossible on the account of next section: in a logic with only stable rules, neither ⊥ nor an arbitrary atomic proposition is provable. A version of Dummett’s conjecture also holds, but it has to be said that it is not the most interesting result.

  4. There are similarities to the view that logic is ‘topic neutral’ or ‘carries no information’.

  5. Prior [32]. Belnap [1] suggests that logical constants need to be conservative, but Prior was not convinced this is sufficient [33].

  6. See Francez and Dyckhoff [13] and Read [36] and literature referred to there.

  7. In an intuitionist modal logic where both □ and ♢ are present, the restrictions may have to be more complicated. For the classical case, we can assume that the logic only has □.

  8. I’m using this restriction as it mirrors the rules of Biermann and de Paiva [2] in the next section. An intuitionist modal logic may require more complicated restrictions, such as those of Prawitz [21].

  9. Besides the works already cited, Dummett’s [5, 6, 9] and [10] contain extensive discussions of the issue.

  10. There are several proposals for how to formulate classical logic in a such a way that its rules may count as harmonious, e.g., Milne [19] and Read [34]. It is fair to say that they all deviate in some way from Dummett’s and Prawitz’ harmony.

  11. Or □A, in modal logic. A local peak with □E after ⊥ E can be levelled.

  12. This argument can be found in Hand [18], but it has probably occurred to many philosophers independently, amongst them the present author.

  13. Tennant [42], which is a response to Hand [18].

  14. Brandom ([3]: Lecture 5, p.8ff). His definition of negation is different from Tennant’s, but there is no need to go into the details here. A similar approach can already be found in Demos [4]. It never really caught on, maybe because of Russell’s criticism ([40]: 5ff), ([41]: 211ff).

  15. See Price [29–31]. Rumfitt [37] develops a formal framework for the account, which has sparked some discussion. See [11, 12, 17, 38, 39]. Textor [43] is a critical assessment of whether there is a speech act of denial that is prior to the assertion of negated sentences. It is worth adding that according to Tennant and Rumfitt, ⊥ isn’t a proposition, but a ‘structural punctuation mark’. Nonetheless, it plays a role in their calculi that can be played by a proposition or speech act. Despite their declarations, someone who is being taught the rules of their calculi may not come to the conclusion that ⊥ is quite so special, and I’m not convinced that would just be a confusion.

  16. Modal notions are almost never discussed in Dummett/Prawitz-style proof-theoretic semantics. Read [35] uses a labelled deductive system, which strikes me as possible worlds semantics dressed up. Pfenning and Davies [20] is closer to what I say here.

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Acknowledgments

I would like to thank Anneli Jefferson, Jessica Leech and Julien Murzi for their very helpful comments on this paper.

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  1. Department of Philosophy, Birkbeck, University of London,, Malet Street, London, WC1E 7HX, UK

    Nils Kürbis

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Kürbis, N. Proof-Theoretic Semantics, a Problem with Negation and Prospects for Modality. J Philos Logic 44, 713–727 (2015). https://doi.org/10.1007/s10992-013-9310-6

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